US7548837B2ExpiredUtilityPatentIndex 51
Simulation of string vibration
Est. expiryJan 14, 2024(expired)· nominal 20-yr term from priority
Inventors:SAPP MARKUS
G10H 5/007
51
PatentIndex Score
0
Cited by
6
References
42
Claims
Abstract
A method of simulating a string using a wave equation which relates movement of the string in time to force acting on the string, wherein the force acting on the string simulates a stream of a fluid medium flowing relative to the string. The simulated string is supported between two supports and is aligned at rest in a first direction between the two supports, a first of which allows movement in a second direction orthogonal to the first direction and a second of which does not allow movement. The string is then caused from rest to vibrate in a plane, which includes the first and second directions, by turbulence in the fluid flow causing the stream of fluid medium to exert a pressure on the string in the second direction. Movement of the string out of alignment with the first direction causes the stream of fluid medium flowing in the first direction to exert a force on the string in the second direction.
Claims
exact text as granted — not AI-modified1. A machine-implemented method for creating a musical sound, the method comprising:
establishing a model of a string that rests in a fluid medium and has a fixed end and a movable end;
simulating a turbulence to excite the moveable end of the string from a rest position;
simulating a force exerted by a stream of the fluid medium flowing in a direction along a longitudinal axis of the string;
calculating a self-sustained vibration of the string in response to the turbulence and the force;
calculating a representation of a sound based on the self-sustained vibration; and
creating the musical sound by a sound generating device based on the representation.
2. The method according to claim 1 , wherein:
the string is supported between two supports and is aligned at rest in a first direction between the two supports;
a first of the two supports allows movement in a second direction orthogonal to the first direction and a second of the two supports does not allow movement; and
the string is caused from rest to vibrate in a plane, which includes the first and second directions, by turbulence in the fluid flow causing the stream of fluid medium to exert a pressure on the string in the second direction.
3. The method according to claim 2 , wherein:
movement of the string out of alignment with the first direction causes the stream of fluid medium flowing in the first direction to exert the force on the string in the second direction.
4. The method according to claim 1 , wherein:
the string is supported between two supports aligned in a first direction,
a first of the two supports is rigid and a second of the two supports allows movement in a second direction orthogonal to the first direction; and
the string is caused to vibrate in a plane, which includes the first and second directions, by the stream of fluid medium flowing in a direction having a component in the second direction.
5. The method according to claim 1 , wherein:
the string is supported between two supports aligned in an x-direction;
a first of the two supports allows movement in a y-direction orthogonal to the x-direction and a second of the two supports does not allow movement;
the string comprises a plurality of discrete elements aligned at rest in the x-direction and spaced apart by a distance dx; and
the discrete elements are able to move in discrete steps of time dt in the y-direction only.
6. The method according to claim 5 , in which the string comprises a plurality of j discrete elements from j=0 at one end movably supported by the first support to j=x−1 at the opposite end immovably supported by the second support; wherein
j is an integer; and
the stream of fluid medium flows in the x-direction and exerts a pressure on the string between elements j=0 and j=1.
7. The method according to claim 6 , wherein the force F PRES [n, 0] at time n acting on the movably supported element j=0 due to the pressure on the string between the movably supported element j=0 and adjacent element j=1 is given by:
F PRES [n, 0]= P *( y[n, 0 ]−y[n, 1])/ dx
in which:
P denotes the pressure exerted by the stream of fluid medium on the string between the movably supported element j=0 and adjacent element j=0;
y[n, 0] denotes the excursion of the movably supported element j=0 at time n; and
y[n, 1] denotes the excursion of the adjacent element j=1 at time n.
8. The method according to claim 6 , wherein the force F TURB [n, 0] at time n acting on the movably supported element j=0 due to the turbulence in the stream of fluid medium is given by:
F TURB[ n, 0]= C TURB *N RND [n]
in which:
C TURB denotes a turbulence coefficient; and
N RND [n] denotes a random signal.
9. The method according to claim 8 , wherein the random signal comprises a low pass filtered noise.
10. The method according to claim 6 , wherein the excursion y[n+1, 0] of the movably supported element for the next sample due to the pressure on the string between the movably supported element j=0 and adjacent element j=1 is given by:
y[n+ 1, 0 ]=y[n, 0]+( F PRES [n, 0])+ F TURB [n, 0])* dt 2/ /M[ 0]
in which:
y[n, 0] denotes the excursion of the movably supported element j=0 at time n; and
F PRES [n, 0] denotes the force at time n acting on the movably supported element j=0 due to the pressure on the string between the movably supported element j=0 and adjacent element j=1;
F TURB [n, 0] denotes the force at time n acting on the movably supported element j=0 due to the turbulence in the stream of fluid medium; and
M[0] denotes the mass of the movably supported element j=0.
11. The method according to claim 10 , wherein the excursion y[n+1, 0] is limited.
12. The method according to claim 6 , wherein the stream of the fluid medium exerts a pressure on the string between each of the elements; and
wherein the force F[n, j] at time n acting on each discrete element from j=1 to j=x−2 due to the pressure P is given by:
F[n, j]=P[j ]*( y[n, j]−y[n, j− 1])/ dx+P[j]* ( y[n, j]−y[n, j+ 1])/ dx.
13. The method according to claim 12 , wherein the pressure P decreases linearly or exponentially with increasing j.
14. The method according to claim 5 , wherein a wave equation is used to relate the movement of the string in time to the excursion of the movable end, the wave equation being an approximation of a continuous wave equation
M
∂
2
y
∂
t
2
=
T
∂
2
y
∂
x
2
-
S
∂
4
y
∂
x
4
+
L
T
∂
3
y
∂
x
2
∂
t
-
L
S
∂
5
y
∂
x
4
∂
t
-
L
v
∂
y
∂
t
+
F
(
x
,
t
)
in which:
F(x, t) denotes an external force at coordinate x on the string at time t;
M denotes mass per length;
S denotes stiffness of the string;
T denotes tension of the string;
Ls denotes a loss associated with the stiffness of the string;
Lt denotes a loss associated with the tension of the string; and
Lv denotes a loss associated with the turbulent flow of the fluid medium.
15. The method according to claim 14 , wherein the approximation of the continuous wave equation is the discrete recursion formula:
y
[
n
+
1
,
j
]
=
(
y
(
n
,
j
-
2
]
·
c
1
+
y
[
n
,
j
-
1
]
·
c
2
+
y
[
n
,
j
]
·
c
3
+
y
[
n
,
j
+
1
]
·
c
2
+
y
[
n
,
j
+
2
]
·
c
1
+
y
[
n
-
1
,
j
-
2
]
·
c
4
+
y
[
n
-
1
,
j
-
1
]
·
c
5
+
y
[
n
-
1
,
j
]
·
c
6
+
y
[
n
1
,
j
+
1
]
·
c
5
+
y
[
n
-
1
,
j
+
2
]
·
c
4
)
/
M
[
j
]
+
2
y
[
n
,
j
]
+
F
[
n
,
j
]
/
M
[
j
]
in which:
dx=1;
dt=1;
y[n, j] denotes the excursion of discrete element j in the y-direction at time n;
y[n 1, j] denotes the excursion of discrete element j in the y-direction at time n+1;
y[n, j+1] denotes the excursion of discrete element j+1 in the y-direction at time n;
M[j] denotes the mass of discrete element j;
F[n, j] denotes an additional external force acting on a discrete element j at time n; and
c1 to c6 are coefficients, which depend on the material parameters of the string and the surrounding media.
16. The method according to claim 15 , wherein
c 1=−( S+Ls );
c 2= T+ 4 S+Lt+ 4 Ls;
c 3=−(2 T+ 6 S+Lv+ 2 Lt+ 6 Ls );
c 4= Ls;
c 5=−( Lt+ 4 Ls ); and
c 6= Lv+ 2 Lt+ 6 Ls.
17. The method according to claim 15 , wherein the discrete recursion formula is solved for the elements adjacent the respective supports by providing a dummy element at opposite ends of the string so that the excursion y[n+1, −1] of a dummy element adjacent the movably supported element for the next sample is given by:
y[n+ 1, −1 ]=y[n+ 1, 0]−( y[n+ 1, 1 ]−y[n+ 1, 0])
and the excursion y[n+1, x] of a dummy element adjacent the immovably supported element for the next sample is given by:
y[n+ 1, x]=−y[n+ 1 , x− 2].
18. A machine readable medium providing executable computer program instructions which when executed cause a data processing system to perform a method for creating a musical sound, the method comprising:
establishing a model of a string that rests in a fluid medium and has a fixed end and a movable end;
simulating a turbulence to excite the moveable end of the string from a rest position;
simulating a force exerted by a stream of the fluid medium flowing in a direction along a longitudinal axis of the string;
calculating a self-sustained vibration of the string in response to the turbulence and the force;
calculating a representation of a sound based on the self-sustained vibration; and
creating the musical sound by a sound generating device based on the representation.
19. The machine readable medium according to claim 18 , wherein:
the string is supported between two supports and is aligned at rest in a first direction between the two supports;
a first of the two supports allows movement in a second direction orthogonal to the first direction and a second of the two supports does not allow movement; and
the string is caused from rest to vibrate in a plane, which includes the first and second directions, by turbulence in the fluid flow causing the stream of fluid medium to exert a pressure on the string in the second direction.
20. The machine readable medium according to claim 19 , wherein:
movement of the string out of alignment with the first direction causes the stream of fluid medium flowing in the first direction to exert the force on the string in the second direction.
21. The machine readable medium according to claim 18 , wherein:
the string is supported between two supports aligned in a first direction,
a first of the two supports is rigid and a second of the two supports allows movement in a second direction orthogonal to the first direction; and
the string is caused to vibrate in a plane, which includes the first and second directions, by the stream of fluid medium flowing in a direction having a component in the second direction.
22. The machine readable medium according to claim 18 , wherein:
the string is supported between two supports aligned in an x-direction;
a first of the two supports allows movement in a y-direction orthogonal to the x-direction and a second of the two supports does not allow movement;
the string comprises a plurality of discrete elements aligned at rest in the x-direction and spaced apart by a distance dx; and
the discrete elements are able to move in discrete steps of time dt in the y-direction only.
23. The machine readable medium according to claim 22 , in which the string comprises a plurality of j discrete elements from j=0 at one end movably supported by the first support to j=x−1 at the opposite end immovably supported by the second support; wherein
j is an integer; and
the stream of fluid medium flows in the x-direction and exerts a pressure on the string between elements j=0 and j=1.
24. The machine readable medium according to claim 23 , wherein the force F PRES [n, 0] at time n acting on the movably supported element j=0 due to the pressure on the string between the movably supported element j=0 and adjacent element j=1 is given by:
F PRES [n, 0]= P* ( y[n, 0 ]−y[n , 1])/ dx
in which:
P denotes the pressure exerted by the stream of fluid medium on the string between the movably supported element j=0 and adjacent element j=0;
y[n, 0] denotes the excursion of the movably supported element j=0 at time n; and
y[n, 1] denotes the excursion of the adjacent element j=1 at time n.
25. The machine readable medium according to claim 23 , wherein the force F TURB [n, 0] at time n acting on the movably supported element j=0 due to the turbulence in the stream of fluid medium is given by:
F TURB [n, 0 ]=C TURB *N RND [n]
in which:
C TURB denotes a turbulence coefficient; and
N RND [n] denotes a random signal.
26. The machine readable medium according to claim 23 , wherein the excursion y[n+1, 0] of the movably supported element for the next sample due to the pressure on the string between the movably supported element j=0 and adjacent element j=1 is given by:
y[n+ 1, 0 ]=y[n , 0]+( F PRES [n, 0]+ F TURB [n, 0])* dt 2 /M [0]
in which:
y[n, 0] denotes the excursion of the movably supported element j=0 at time n; and
F PRES [n, 0] denotes the force at time n acting on the movably supported element j=0 due to the pressure on the string between the movably supported element j=0 and adjacent element j=1;
F TURB [n, 0] denotes the force at time n acting on the movably supported element j=0 due to the turbulence in the stream of fluid medium; and
M[0] denotes the mass of the movably supported element j=0.
27. The machine readable medium according to claim 26 , wherein the excursion y[n+1, 0] is limited.
28. The machine readable medium according to claim 23 , wherein the stream of the fluid medium exerts a pressure on the string between each of the elements; and
wherein the force F[n, j]at time n acting on each discrete element from j=1 to j=x−2 due to the pressure P is given by:
F[n, j]=P[j]* ( y[n, j]−y[n, j− 1])/ dx+P[j]* ( y[n, j+ 1])/ dx.
29. The machine readable medium according to claim 28 , wherein the pressure P decreases linearly or exponentially with increasing j.
30. The machine readable medium according to claim 22 , wherein a wave equation is used to relate the movement of the string in time to the excursion of the movable end, the wave equation being an approximation of a continuous wave equation
M
∂
2
y
∂
t
2
=
T
∂
2
y
∂
x
2
-
S
∂
4
y
∂
x
4
+
L
T
∂
3
y
∂
x
2
∂
t
-
L
S
∂
5
y
∂
x
4
∂
t
-
L
v
∂
y
∂
t
+
F
(
x
,
t
)
in which:
F(x, t) denotes an external force at coordinate x on the string at time t;
M denotes mass per length;
S denotes stiffness of the string;
T denotes tension of the string;
Ls denotes a loss associated with the stiffness of the string;
Lt denotes a loss associated with the tension of the string; and
Lv denotes a loss associated with the turbulent flow of the fluid medium.
31. The machine readable medium according to claim 30 , wherein the approximation of the continuous wave equation is the discrete recursion formula:
y
[
n
+
1
,
j
]
=
(
y
[
n
,
j
-
2
]
c
1
+
y
[
n
,
j
-
1
]
c
2
+
y
[
n
,
j
]
c
3
+
y
[
n
,
j
+
1
]
c
2
+
y
[
n
,
j
+
2
]
c
1
+
y
[
n
-
1
,
j
-
2
]
c
4
+
y
[
n
-
1
,
j
-
1
]
c
5
+
y
[
n
-
1
,
j
]
c
6
+
y
[
n
1
,
j
+
1
]
c
5
+
y
[
n
-
1
,
j
+
2
]
c
4
)
/
M
[
j
]
+
2
y
[
n
,
j
]
+
F
[
n
,
j
]
/
M
[
j
]
in which:
dx=1;
dt=1;
y[n, j] denotes the excursion of discrete element j in the y-direction at time n;
y[n+1, j] denotes the excursion of discrete element j in the y-direction at time n+1;
y[n, j+1] denotes the excursion of discrete element j+1 in the y-direction at time n;
M[j] denotes the mass of discrete element j;
F[n, j] denotes an additional external force acting on a discrete element j at time n; and
c1 to c6 are coefficients, which depend on the material parameters of the string and the surrounding media.
32. The machine readable medium according to claim 31 , wherein
c 1=−( S+Ls );
c2= T+ 4 S+Lt+ 4 Ls;
c 3=−(2 T+ 6 S+Lv+ 2 Lt+ 6 Ls );
c 4= Ls;
c 5=−( Lt+ 4 Ls ); and
c 6= Lv+ 2 Lt+ 6 Ls.
33. The machine readable medium according to claim 31 , wherein the discrete recursion formula is solved for the elements adjacent the respective supports by providing a dummy element at opposite ends of the string so that the excursion y[n+1, −1] of a dummy element adjacent the movably supported element for the next sample is given by:
y[n+ 1, −1 ]=y[n+ 1, 0]−( y[n+ 1, 1 ]−y[n+ 1, 0])
and the excursion y[n+1, x] of a dummy element adjacent the immovably supported element for the next sample is given by:
y[n +1 , x]=−y[n +1 , x −2].
34. An apparatus comprising:
a processing element to establish a model of a string that rests in a fluid medium and has a fixed end and a movable end, to simulate a turbulence to excite the moveable end of the string from a rest position, to simulate a force exerted by a stream of the fluid medium flowing in a direction along a longitudinal axis of the string, the processing element further to calculate a self-sustained vibration of the string in response to the turbulence and the force, and to calculate a representation of a sound based on the self-sustained vibration
a sound generating element, coupled to the processing element, to create a musical sound based on the representation; and
a storage device, coupled to the processing element, to store data used by the processing elements.
35. The apparatus according to claim 34 , wherein:
the string is supported between two supports and is aligned at rest in a first direction between the two supports;
a first of the two supports allows movement in a second direction orthogonal to the first direction and a second of the two supports does not allow movement; and
the string is caused from rest to vibrate in a plane, which includes the first and second directions, by turbulence in the fluid flow causing the stream of fluid medium to exert a pressure on the string in the second direction.
36. The apparatus according to claim 35 , wherein:
movement of the string out of alignment with the first direction causes the stream of fluid medium flowing in the first direction to exert the force on the string in the second direction.
37. The apparatus according to claim 34 , wherein:
the string is supported between two supports aligned in a first direction,
a first of the two supports is rigid and a second of the two supports allows movement in a second direction orthogonal to the first direction; and
the string is caused to vibrate in a plane, which includes the first and second directions, by the stream of fluid medium flowing in a direction having a component in the second direction.
38. The apparatus according to claim 34 , wherein:
the string is supported between two supports aligned in an x-direction;
a first of the two supports allows movement in a y-direction orthogonal to the x-direction and a second of the two supports does not allow movement;
the string comprises a plurality of discrete elements aligned at rest in the x-direction and spaced apart by a distance dx; and
the discrete elements are able to move in discrete steps of time dt in the y-direction only.
39. The apparatus according to claim 38 , in which the string comprises a plurality of j discrete elements from j=0 at one end movably supported by the first support to j=x−1 at the opposite end immovably supported by the second support; wherein
j is an integer; and
the stream of fluid medium flows in the x-direction and exerts a pressure on the string between elements j=0 and j=1.
40. The apparatus according to claim 39 , wherein the force F PRES [n, 0 ]at time n acting on the movably supported element j=0 due to the pressure on the string between the movably supported element j=0 and adjacent element j=1 is given by:
F PRES [n, 0 ]=P *( y[n, 0 ]−y[n, 1])/ dx
in which:
P denotes the pressure exerted by the stream of fluid medium on the string between the movably supported element j=0 and adjacent element j=0;
y[n, 0] denotes the excursion of the movably supported element j=0 at time n; and
y[n, 1] denotes the excursion of the adjacent element j=1 at time n.
41. The apparatus according to claim 39 , wherein the force F TURB [n, 0]at time n acting on the movably supported element j=0 due to the turbulence in the stream of fluid medium is given by:
F TURB [n, 0]= C TURB *N RND [n]
in which:
C TURB denotes a turbulence coefficient; and
N RND [n] denotes a random signal.
42. The apparatus according to claim 39 , wherein the excursion y[n+1, 0]of the movably supported element for the next sample due to the pressure on the string between the movably supported element j=0 and adjacent element j=1 is given by:
y[n+ 1, 0]= y[n, 0]+( F PRES [n, 0 ]+F TURB [n, 0])* dt 2 /M[ 0]
in which:
y[n, 0] denotes the excursion of the movably supported element j=0 at time n; and
F PRES [n, 0] denotes the force at time n acting on the movably supported element j=0 due to the pressure on the string between the movably supported element j=0 and adjacent element j=1;
F TURB [n, 0]denotes the force at time n acting on the movably supported element j=0 due to the turbulence in the stream of fluid medium; and
M[0] denotes the mass of the movably supported element j=0.Cited by (0)
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